A model for micro-scale propulsion using flexible rotating flagella

Micro-scale propulsion by rotating helical flagella is of interest for the study of bacteria and robotic micro- swimmers. The propulsive thrust and torque produced by the rotating flagella are usually estimated assuming that they are rigid. In this paper we assume the flagella to be deformable elastic rods and compute propulsive forces and torques by enforcing local equilibrium of the rod within the context of resistive force theory. The torque-speed characteristics of the flagellar motor driving the rotation are taken into account. We show that the problem can be cast as a system of algebraic equations if the flagella are assumed to be helical before and after deformation when no spontaneous curvature is included. If the assumption of helical shape is dropped then we show that the propulsion problem can be cast as a system of first order differential equations that can be solved numerically. Our results in both cases agree reasonably well with experimental observations of bacterial propulsion and deviate from the predictions of Purcell depending on the mechanical properties of the flagellum.